scholarly journals On the Brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid

1928 ◽  
Vol 119 (781) ◽  
pp. 34-54 ◽  
Keyword(s):  
Distribution Function ◽  
Equilibrium Distribution ◽  
Mean Free Path ◽  
Physical Analysis ◽  
Brownian Particles ◽  
Uniform Case ◽  
The Mean ◽  
The One ◽  
Coefficient Of Diffusion ◽  
Uniform Fluid

1. In two famous papers on the Brownian motion of grains suspended in a stationary uniform fluid (liquid or gaseous) Einstein obtained, inter alia , the distribution function for the displacements of the grains during any interval t = 0 to t = τ from their positions at time t = 0. The object of this paper is to determine the distribution function for the more general case of a non-uniform fluid. The non-uniformity may refer to temperature, composition, or any other property which affects the coefficient of diffusion (D) of the grains in the fluid. The distribution function is given in 8, where it is shown how its accuracy might be experimentally tested. It contains terms additional to the one given by Einstein for the uniform case; certain of these are definitely determined, but another important term contains a coefficient that cannot be evaluated by considerations of the kind used in this paper (which depend purely on the conservation of the number of grains), but requires more detailed physical analysis; it is surmised that this coefficient vanishes in the case of Brownian particles which are large compared with the mean free path of the surrounding molecules. The main results of the paper refer to the rate of diffusion of the grains due to the non-uniformity of the fluid (9), and to the equilibrium distribution of the grains (10, 11). It is found that if their density is the same as that of the fluid, so that there is no tendency for them to settle in the lower strata, their steady distribution when the fluid is non-uniform is such that the concentration n (or number per unit volume of the fluid) is inversely proportional to D; a solution is also given for the case when the densities are not equal.

Transport processes in dilute gases over the whole range of Knudsen numbers. Part 1. General theory

1979 ◽  
Vol 93 (3) ◽  
pp. 585-607 ◽  
Author(s):  
L. C. Woods
Keyword(s):  
Free Path ◽  
Mean Free Path ◽  
Transport Processes ◽  
Burnett Equations ◽  
Flux Vector ◽  
Dilute Gases ◽  
Knudsen Numbers ◽  
Heat Flux Vector ◽  
The Mean ◽  
The One

The mean-free-path approach to kinetic theory, initiated by Maxwell, and largely abandoned after the Chapman-Enskog success with Boltzmann's equation, is revised and considerably extended in order to find expressions for the heat flux vector q and pressure tensor p, valid (it is hoped) for all Knudsen numbers, K. These expressions (equations (2.24) and (2.26)) are integrals taken over the whole volume of the fluid plus surface integrals taken over the solid boundaries. The one phenomenological element is the mean free path λ, which takes different values according to whether it is mass, momentum or energy that is transported by the molecules. The need for such an approach is evidenced by the existence of critical values of K, above which the Chapman-Enskog expansion in powers of K, truncated after a finite number of terms, fails to yield a solution. For example with the Burnett equations, which are correct to O(K2), the critical K in a shock wave is only 0·2 based upon the upstream λ.

Poiseuille Flow and Thermal Transpiration of a Rarefied Gas Between Parallel Plates: Effect of Nonuniform Surface Properties of the Plates in the Transverse Direction

2015 ◽  
Vol 137 (10) ◽  
Author(s):  
Toshiyuki Doi
Keyword(s):  
Distribution Function ◽  
Flow Rate ◽  
Poiseuille Flow ◽  
Transverse Direction ◽  
Rarefied Gas ◽  
Mean Free Path ◽  
Accommodation Coefficient ◽  
Parallel Plates ◽  
Nonuniform Surface ◽  
The Mean

Poiseuille flow and thermal transpiration of a rarefied gas between parallel plates with nonuniform surface properties in the transverse direction are studied based on kinetic theory. We considered a simplified model in which one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary on which the accommodation coefficient varies periodically and smoothly in the transverse direction. The spatially two-dimensional (2D) problem in the cross section is studied numerically based on the linearized Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The flow behavior, i.e., the macroscopic flow velocity and the mass flow rate of the gas as well as the velocity distribution function, is studied over a wide range of the mean free path of the gas and the parameters of the distribution of the accommodation coefficient. The mass flow rate of the gas is approximated by a simple formula consisting of the data of the spatially one-dimensional (1D) problems. When the mean free path is large, the distribution function assumes a wavy variation in the molecular velocity space due to the effect of a nonuniform surface property of the plate.

Solution with Third-order Accuracy for the Electron Distribution Function in Weakly Ionized Gases (or in Intrinsic Semiconductors): Application to the Drift Velocity

1981 ◽  
Vol 34 (4) ◽  
pp. 361 ◽  
Author(s):  
G Cavalleri
Keyword(s):  
Distribution Function ◽  
Legendre Polynomials ◽  
Taylor Expansion ◽  
Electron Distribution ◽  
Electron Distribution Function ◽  
Mean Free Path ◽  
Third Order ◽  
Order Accuracy ◽  
Weakly Ionized ◽  
The Mean

The first four components 10, I" 12 and 13 of the expansion in Legendre polynomials of the electron distribution function I are shown to be of order t:D, et, e2 and e3 respectively, with e = (m/M)'/2 where m and M are the masses of the electron and molecule respectively. This allows the solution of the so-called P3 approximation to the Boltzmann equation applied to a weakly ionized gas (or to an intrinsic semiconductor) in steady-state and uniform conditions and for dominant elastic collisions. However, nonphysical divergences appear in 10 and in the drift velocity W. This can be understood by the equivalence of the Boltzmann-Legendre formulation and the mean free path formulation in which a Taylor expansion is performed around the 'origin', i.e. for a -+ 0, where a = eE/m is the acceleration due to an external electric field E. Indeed, one sees that the expansion under the integral sign (integrals appear in the evaluation of transport quantities) leads to divergent integrals if the expansion is around a = O. Fortunately, it is easy to perform a Taylor expansion around a oft 0 in the mean free path formulation and then to find the corresponding expansion in Legendre polynomials outside the origin. In this way, explicit convergent expressions are found for 10, I" 12, 13 and W, with third-order accuracy in e = (m/M)'/2. This is better than the best preceding expression, that by Davydov-Chapman-Cowling, which has first-order accuracy only (it is the solution of the P, approximation to the Boltzmann equation).

scholarly journals Departure from MHD prescriptions in shock formation over a guiding magnetic field

2017 ◽  
Vol 35 (3) ◽  
pp. 513-519 ◽  
Author(s):  
A. Bret ◽  
A. Pe'er ◽  
L. Sironi ◽  
M.E. Dieckmann ◽  
R. Narayan
Keyword(s):  
Magnetic Field ◽  
Free Path ◽  
Mean Free Path ◽  
Distribution Functions ◽  
Shock Formation ◽  
Physical Analysis ◽  
Particle In Cell ◽  
The Mean ◽  
Collective Plasma ◽  
Aligned Magnetic Field

AbstractIn plasmas where the mean-free-path is much larger than the size of the system, shock waves can arise with a front much shorter than the mean-free-path. These so-called “collisionless shocks” are mediated by collective plasma interactions. Studies conducted so far on these shocks found that although binary collisions are absent, the distribution functions are thermalized downstream by scattering on the fields, so that magnetohydrodynamics prescriptions may apply. Here we show a clear departure from this pattern in the case of Weibel shocks forming over a flow-aligned magnetic field. A micro-physical analysis of the particle motion in the Weibel filaments shows how they become unable to trap the flow in the presence of too strong a field, inhibiting the mechanism of shock formation. Particle-in-cell simulations confirm these results.

scholarly journals Theory of the formation of a collisionless Weibel shock: pair vs. electron/proton plasmas

2016 ◽  
Vol 34 (2) ◽  
pp. 362-367 ◽  
Author(s):  
A. Bret ◽  
A. Stockem Novo ◽  
R. Narayan ◽  
C. Ruyer ◽  
M. E. Dieckmann ◽  
...  
Keyword(s):  
Mean Free Path ◽  
Particle Accelerators ◽  
Incoming Flow ◽  
Collisionless Shock ◽  
Collisionless Shocks ◽  
Trapping Time ◽  
Particle In Cell ◽  
The Mean ◽  
The One ◽  
Overlapping Region

AbstractCollisionless shocks are shocks in which the mean-free path is much larger than the shock front. They are ubiquitous in astrophysics and the object of much current attention as they are known to be excellent particle accelerators that could be the key to the cosmic rays enigma. While the scenario leading to the formation of a fluid shock is well known, less is known about the formation of a collisionless shock. We present theoretical and numerical results on the formation of such shocks when two relativistic and symmetric plasma shells (pair or electron/proton) collide. As the two shells start to interpenetrate, the overlapping region turns Weibel unstable. A key concept is the one of trapping time τp, which is the time when the turbulence in the central region has grown enough to trap the incoming flow. For the pair case, this time is simply the saturation time of the Weibel instability. For the electron/proton case, the filaments resulting from the growth of the electronic and protonic Weibel instabilities, need to grow further for the trapping time to be reached. In either case, the shock formation time is 2τp in two-dimensional (2D), and 3τp in 3D. Our results are successfully checked by particle-in-cell simulations and may help designing experiments aiming at producing such shocks in the laboratory.

scholarly journals Кинетическая теория пристеночного слоя при произвольных условиях в газоразрядной плазме

2019 ◽  
Vol 89 (9) ◽  
pp. 1384
Author(s):  
O. Мурильо ◽  
А.С. Мустафаев ◽  
В.С. Сухомлинов
Keyword(s):  
Distribution Function ◽  
Electron Energy ◽  
Electron Distribution Function ◽  
Debye Length ◽  
Ion Distribution ◽  
Ion Energy ◽  
Charge Exchange Cross Section ◽  
The Mean ◽  
Exchange Cross Section ◽  
The One

For an arbitrary relation of the Debye length to the ion free path, based on the kinetic approach, it was solved the self consistent problem of the structure of the perturbed wall sheath in a DC plasma gas discharge, near a flat surface at a negative potential in relation to the plasma. The solution was obtained without an artificial separation of the perturbed wall sheath into a quasineutral "presheath" and a wall sheath where quasineutrality is substantially violated. They were taken into account the real ion distribution function in the unperturbed plasma, the dependence of the charge exchange cross section on the ion energy, and the nonzero electric field in the unperturbed plasma. It was found that, if the mean electron energy is preserved, the structure of the perturbed wall sheath weakly depends on the form of the electron distribution function. It was shown that, even under the assumption that the mean electron energy is much higher than the mean ion energy, is the mean ion energy in the unperturbed plasma the one that has a significant effect both on the structure of the quasineutral "presheath" and on the structure of a part of the wall sheath, where there is no quasineutrality. The calculations of the parameters of the ion flux and the structure of the perturbed wall sheath are in agreement with other authors’ experimental results that didn’t have previously an adequate interpretation.

scholarly journals Influence of Shape Spread in an Ensemble of Metal Nanoparticles on Their Optical Properties

2018 ◽  
Vol 63 (3) ◽  
pp. 204 ◽  
Author(s):  
P. M. Tomchuk ◽  
V. N. Starkov
Keyword(s):  
Distribution Function ◽  
Metal Nanoparticles ◽  
Theoretical Basis ◽  
Mean Free Path ◽  
Spherical Nanoparticles ◽  
Shape Distribution ◽  
Nanoparticle Shape ◽  
Joint Application ◽  
Tensor Quantity ◽  
The Mean

The theoretical basis of the work consists in that the dissipative processes in non-spherical nanoparticles, whose sizes are smaller than the mean free path of electrons, are characterized by a tensor quantity, whose diagonal elements together with the depolarization coefficients determine the half-widths of plasma resonances. Accordingly, the averaged characteristics are obtained for an ensemble of metal nanoparticles with regard for the influence of the nanoparticle shape on the depolarization coefficients and the components of the optical conductivity tensor. Three original variants of the nanoparticle shape distribution function are proposed on the basis of the joint application of the Gauss and “cap” functions.

T Torque on a Small Particle in a Nonhomogeneous Monatomic Gas

1981 ◽  
Vol 36 (6) ◽  
pp. 559-567
Author(s):  
H. Vestner ◽  
J. Halbritter
Keyword(s):  
Boundary Condition ◽  
Distribution Function ◽  
Velocity Distribution ◽  
Small Particle ◽  
Velocity Distribution Function ◽  
Mean Free Path ◽  
Pressure Tensor ◽  
The Mean ◽  
Geometrical Factors ◽  
Monatomic Gas

Abstract The torque a particle experiences in a monatomic gas in non-equilibrium is calculated with the method Waldmann originally used to compute the force. For a particle which is much smaller than the mean free path in the gas this method involves the following two essential steps: i) with the help of a boundary condition the torque is expressed by the velocity distribution function of the gas atoms approaching the particle; ii) this undisturbed distribution function is approximated by a moments-expansion, restricted here to thirteen moments. In the resulting expression for the torque the pressure tensor and the local vorticity in the gas occur. The geometrical factors are given for particles shaped as spheroids, cylinders and spherocylinders

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